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Fortschritte der Physik

Dynamics of a single trapped ion in an optical underdamped cavity

Sabrina Maniscalco, A. Messina, C. Di Fidio, W. Vogel

Published Feb 26, 2003• 1 citation
Publisher

Abstract

Abstract The dynamics of a single trapped ion placed inside a high Q optical cavity is studied in presence of cavity losses and far from the Lamb‐Dicke regime. In the underdamped cavity limit, analytical results for describing the dynamics of the system are derived making use of the secular approximation. Our method allows to obtain analytical expressions for the time evolution of the joint vibration‐photon number distribution and for the occupation probability of the upper electronic state of the ion.

DOI
10.1002/prop.200310024

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Abstract The dynamics of a single trapped ion placed inside a high Q optical cavity is studied in presence of cavity losses and far from the Lamb‐Dicke regime. In the underdamped cavity limit, analytical results for describing the dynamics of the system are derived making use of the secular approximation. Our method allows to obtain analytical expressions for the time evolution of the joint vibration‐photon number distribution and for the occupation probability of the upper electronic state of the ion.
The dynamics of the interaction of a quantized cavity field and the vibronic degrees of freedom of a trapped ion is studied under realistic conditions by including cavity losses, spontaneous electronic transitions, and atomic nonlinearities. As long as spontaneous electronic transitions are negligible, analytical results are derived for describing the interaction of the trapped ion and the damped cavity field in the secular approximation. Under more general conditions, when the secular approximation breaks down and spontaneous emission effects become important, the dynamics of the system is studied by quantum-trajectory methods. As an example we demonstrate that, by exploiting the nonlinearities inherent in the dynamics of the system, one may prepare correlated vibronic states and vibrational Fock states. Moreover, we study the generation of an entangled Greenberger-Horne-Zeilinger state. Taking into account the decoherence due to electronic and cavity losses, the best fidelity is obtained under conditions far from the Lamb-Dicke regime.
In this paper, we generalize the ordinary two-photon Jaynes-Cummings model (TPJCM) by considering the atom (or ion) to be trapped in a simple harmonic well. A typical setup would be an optical cavity containing a single ion in a Paul trap. Due to the inclusion of atomic vibrational motion, the atom-field coupling becomes highly nonlinear what brings out quite different behaviors for the system dynamics when compared to the ordinary TPJCM. In particular, we derive an effective two-photon Hamiltonian with dependence on the number operator of the ion's center-of-mass motion. This dependence occurs both in the cavity induced Stark-shifs and in the ion-field coupling, and its role in the dynamics is illustrated by showing the time evolution of the probability of occupation of the electronic levels for simple initial preparations of the state of the system.
We analyze the quantum motion of a cold, trapped two-level ion interacting with a quantized light field in a single-mode cavity. We show that in the nonclassical Lamb-Dicke limit the time evolution of the vibrational mode representing the quantized motion of the center of mass of the trapped ion is very sensitive to the quantum statistics of the light field. We also show that the system under consideration may evolve into the maximally entangled three-particle Greenberger-Horne-Zeilinger state. We briefly discuss the dynamics of a cluster of two-level ions trapped in a cavity and interacting with a quantized light field.
We investigate the dynamics of cavity cooling of a single ion beyond the Lamb-Dicke regime and demonstrate a cooling limit of approximately 50% of the Doppler temperature using a high finesse cavity. We also study the statistical properties of the ion-cavity emission and present a model that maps the phonon states to the photon states. With this model, we explain the super-Poissonian photon distribution observed in our experiments and propose a method to estimate the ion temperature in a real-time manner based on the statistical behavior of the photon emission from the cavity.
We investigate the dynamics of a cold trapped ion coupled to the quantized field inside a high-finesse cavity, considering exact resonance between the ionic internal levels and the field (carrier transition). We derive an intensity-dependent Hamiltonian in which terms proportional to the square of the Lamb-Dicke parameter $(\ensuremath{\eta})$ are retained. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion and field and on the values of $\ensuremath{\eta}.$
We propose a model for a single-trapped-ion vibronic Raman laser and study its dynamics by using quantum-trajectory methods. In our treatment, it is essential that both the cavity field of the high-finesse optical cavity and the center-of-mass vibrational motion of the trapped ion be quantized. A transition from a super-Poissonian light source to a Poissonian lasing regime is obtained by increasing the Raman coupling constant. Furthermore, we demonstrate that a nonclassical regime can be realized, where the photon statistics becomes sub-Poissonian and the photons leak out of the cavity in an antibunched manner. This is achieved by exploiting nonlinear Stark shifts inherent in the model, which depend on both the number of cavity photons and the number of vibrational quanta.